Optimization of jamming formation of USV offboard active decoy clusters based on an improved PSO algorithm

Zhaodong Wu, Yasong Luo, Shengliang Hu

College of Weapons Engineering, PLA Naval University of Engineering, Wuhan, 430033, China

Keywords:Electronic countermeasure Offboard active decoy USV cluster Jamming formation optimization Improved PSO algorithm

ABSTRACT Offboard active decoys (OADs) can effectively jam monopulse radars.However, for missiles approaching from a particular direction and distance, the OAD should be placed at a specific location, posing high requirements for timing and deployment.To improve the response speed and jamming effect,a cluster of OADs based on an unmanned surface vehicle(USV)is proposed.The formation of the cluster determines the effectiveness of jamming.First, based on the mechanism of OAD jamming, critical conditions are identified, and a method for assessing the jamming effect is proposed.Then, for the optimization of the cluster formation,a mathematical model is built,and a multi-tribe adaptive particle swarm optimization algorithm based on mutation strategy and Metropolis criterion (3M-APSO) is designed.Finally, the formation optimization problem is solved and analyzed using the 3M-APSO algorithm under specific scenarios.The results show that the improved algorithm has a faster convergence rate and superior performance as compared to the standard Adaptive-PSO algorithm.Compared with a single OAD, the optimal formation of USV-OAD cluster effectively fills the blind area and maximizes the use of jamming resources.

1.Introduction

As one concrete application of off-platform active jamming technology on naval battlefields,offboard active decoys(OADs)can effectively interfere with the monopulse radar of anti-ship missiles(ASM) [1].An OAD functions by transmitting radar signals and is deployed at a specific location relative to the protected vessel to cause angle-positioning deviations to the radar seeker [2,3].

Recently, off-platform radar active decoys have received increasing attention.A variety of jamming strategies have been proposed for different combat scenarios.Depending on their deployment characteristics, active decoys can be categorized into fixed, towed, buoyed, parachuted, hovered decoys, and drones.In land-based anti-missile operations, active decoys are primarily used to jam anti-radiation missiles and are typically deployed around protected radar stations.To achieve a better jamming effect,active decoys and protected platforms generally form a"triangle"or"diamond"shape[4,5].Most air platforms carry towed or launched decoys.Towed decoys have a towline between 90 m and 120 m[6].When the missile is traveling at 0.8 Mach,the aircraft has a survival probability of 80%after launching the decoys 2 km from the missile and then turning to maneuver [7,8].In maritime anti-missile operations, active decoys are mostly floating, hovering, and parachuting ones.The“triangle”jamming formation of the decoy,ship and missile was quantitatively described in analogy with the chaff[9].Using"Nulka"as an example,the jamming process of OAD can be typically divided into four stages,i.e.,the ship being targeted,the angle deviated,the range gate captured and the ship escaping[10].From an energy perspective, the jamming-to-signal ratio (JSR)should be above 5 dB.If the direction of arrival (DOA) is 0°, the decoy should be deployed at 70°-110°from the ship bow,and the distance should be maintained at 500 m [11,12].The forwarding delay of the jammer was further considered, and a forward-out deployment distance for the decoy was proposed [13].

Based on these studies,the position of the decoy relative to the protected platform is crucial for effective jamming.Its position decides the conical jamming blind zone [14], meaning that the decoy can only interfere with missiles coming from particular directions and distances.In land-based anti-missile operations,more than one decoy is typically employed to meet the jamming requirements of different DOAs[4,5].For airborne platforms,an airlaunched decoy can obtain an effective jamming situation within a very short time by moving rapidly relative to the aircraft [7,15].Unlike land- and air-based platforms, the protected ship is a moving platform in maritime anti-missile operations, yet it moves relatively slowly compared with the missile.It takes more time for OADs to form an effective jamming situation dynamically.

Compared with studies on air platforms and land-based decoy jamming strategies, studies on maritime OAD have defined extra prerequisite constraints on the DOA of the missile.Although a parachuting or hovering decoy can be launched from a ship and create a dynamic jamming situation,it only has a limited duration and is easily affected by sea winds and waves.Therefore, it is challenging to guarantee a ship's interference capability against missiles, and no suitable solution has been proposed to solve the jamming blindness problem of OADs in maritime applications.

Recently, unmanned surface vehicles (USVs) have been developing rapidly.In the military field, with the breakthroughs and applications of autonomous sensing, path planning, intelligent decision-making and other related technologies [16-18], USV has become an essential extension of maritime combat forces.NATO countries have used USV as the carrier of OAD and conducted relevant experiments [19-21], as shown in Fig.1.Compared with parachuting, hovering and UAV decoys, USV-based OADs (USVOADs) are highly integrated and have good maneuverability, long working duration and high adaptability to maritime environments[22-24].They can accompany a protected vessel for a long time and are reliable in maritime anti-missile operations.

Inspired by the USV cluster used for jamming proposed in Ref.[25], this study proposes using USV-OADs to form an unmanned cluster,overcome the OAD's limitations and maximize the jamming effect.Some highlights of our paper are as follows.Firstly,to objectively describe the effect of the jamming cluster, we propose the assessment method from the perspective of the interference mechanism.Secondly,we establish the multi-decoy formation optimization model, in which the evaluation index, jamming constraints,and parameters of USV formation are considered.Thirdly,after analyzing the characteristic of this model, we introduce the 3M-APSO algorithm, of which the improved strategies make the algorithm suitable for the high-dimensional USV-OADs formation optimization problem.Finally, simulations for different scenarios verify the effectiveness of the algorithm and help guide the application strategy of the USV-OAD cluster.

This paper is organized as follows.Section 2 analyzes critical conditions for effective jamming based on the mechanism of OAD jamming and proposes a method for assessing the effectiveness of jamming based on probabilistic reasoning.Section 3 develops an OAD cluster formation optimization model and designs a multitribe adaptive particle swarm optimization (PSO) algorithm based on the mutation strategy and Metropolis criterion(3M-APSO).The optimization is then realized using the improved algorithm.Section 4 verifies the optimization performance of the algorithm through numerical simulation analysis under different defense requirements and determines the optimized formation using theories.The conclusions are summarized in Section 5.

Fig.1.Different OAD equipment: (a) "Nulka" hovering OAD; (b) "Haylcon" USV of NATO equipped with OAD.

2.Analysis of the OAD jamming effectiveness

2.1.Jamming mechanism

For the same beamwidth and range gate, in the presence of OADs, the output of the radar seeker is the result of the noncoherent synthesis of the ship echo signal and the decoy jamming signal.For a single OAD, based on the sum-difference method of amplitude and phase, the radar seeker points to the synthesized centroid of the decoy and ship echo complex voltages,weighted by the beam gain [26], as follows:

where a0and a1indicate the voltage amplitudes of the ship's echo signal and decoy's interference signal, respectively.φ1, a random variable, indicates the phase difference between two signals.S(θ)indicates the antenna direction gain.θ0and θ1indicate the azimuth angles of the ship and the OAD relative to the beam axis of the radar, respectively.

If multiple decoys interfere with the missile simultaneously, as shown in Fig.2, according to Eqs.(1) and (2), the angle can be calculated as follows:

where N is the number of decoys and φirepresents the phase difference between the jamming signal of each decoy and the ship's echo signal.

As φiis random,θmis also a random variable.Suppose that the phase difference φiis uniformly distributed between[0,2π].In that case, the conventional average of the radar-indicated angle can be considered as the orientation of the signal source with a higher power.When the amplitude of each measured signal is used as the weighted value of the single goniometry, the average indicated angle is the power center of mass of each source[27],expressed as follows:

Fig.2.Jamming of multiple OADs.

As JSR is typically greater than 1, the deviation of the weighted average caused by OAD can be observed to be smaller than that of the conventional average by comparing the weighted average with the conventional average.To obtain a more robust jamming effect,this study adopts the optimization strategy of USV-OAD formation mainly based on the weighted average value of the indicated angle.

2.2.Critical conditions for effective jamming

After the anti-ship missile locks onto the target, the primary function of the OAD is to cause angular deception to the radar seeker,allowing the ship to escape the tracking beam.In this way,even if the missile re-locates and locks onto the target once again,it will be off-target due to steering overload.Due to the dynamic nature of missile maneuvering,the jamming effect of the OAD also changes constantly.The critical jamming scenario for a single OAD is shown in Fig.3.

As the missile approaches,the angle between the decoy and the ship relative to the missile increases continuously.According to Fig.3,at the time t+1,the decoy and the ship are on the edge of the tracking beam.According to the radar antenna pattern[28,29],the antenna gain at the edge changes fast.It indicates that after the time t+1, regardless of which side is pointed by the radar seeker,the other side will be able to escape the tracking beam rapidly.

According to Eq.(2), the jamming and ship echo signals are amplitude-modulated by the antenna directive gain.The echo signal of the ship is modulated twice as a passive radiation source.According to the radar and radar jamming equations [30], the voltage of the jamming and ship echo signals can be expressed as follows:

These parameters are defined in Table 1.Gtis a function of the angle between the beam axis and the ship or jammer.As for critical conditions,the radar seeker indicates an angle of 0°,and the decoy and ship are located at the edge of the beam, angles of which are half of the beamwidth.The following equations hold.

Fig.3.Critical jamming scenario of a single OAD.

Table 1 Parameters of offboard active decoy jamming.

Substituting Eqs.(5), (7) into Eq.(1), we obtain the critical condition for the missile distance.

As shown in Eqs.(5) and (6), b1is inversely proportional to R,and b0is inversely proportional to R squared.At the time t in Fig.3,if R>Rc,then b1>b0,indicating the radar seeker points to the OAD side.Conversely,if R

According to Fig.2, the critical condition for multiple OADs is that the outermost active decoy and ship are on the edge of the tracking beam.Similar to the aforementioned derivation, critical missile distance for multiple OADs can be expressed as follows:

As observed,Rcis related to the deployment of other decoys.In the case of OAD 2 in Fig.2,when the location of OAD 2 is known,θ2changes with Rc.According to Eq.(9), if θ2<0 where OAD 1 and OAD 2 are on different sides, Rcobtained by only using OAD 1 is smaller than that obtained by using both OAD 1 and OAD 2, indicating OAD 2 has a negative effect on jamming.When θ2>0 with OAD 2 and OAD 1 on the same side, OAD 2 amplifies the jamming effect.According to the geometric relationship, the distance between the OAD and the ship under the critical condition, denoted as Yc,can be calculated as follows:

2.3.Method for assessing OAD jamming effect

The effect of decoy placement can be quantitatively evaluated by assessing the jamming effectiveness.Based on the analysis in subsection 2.2, we can infer whether the OAD placement is reasonable given the radar seeker's parameters and a stable jamming environment.However,in practice, the parameters are often unknown, and the jamming process is subject to numerous uncertainties.

According to the critical conditions in Eqs.(8)and(13),the main unknowns include the transmit power of the radar seeker, beamwidth, antenna directive gain, radar cross-section of the ship and jamming loss.Consequently, the unknown elements can be expressed quantitatively in terms of probabilities, and the probability distribution of the critical conditions can be calculated based on probabilistic inference methods, providing an assessment method.

The theorem used is as follows.

where p1denotes the probability of a jamming signal entering the radar seeker.Pvcan be described as the effective jamming probability.However, the missile distance R is not a random variable.It mainly reflects the proportion of the missile's jammable distance to the overall terminal guidance distance.Let Rtdenote the distance at which the missile starts tracking,and

When multiple decoys interfere with the incoming missile simultaneously,the probability P(Y1>Yc)is improved.In addition,multiple decoys can introduce an additional jamming gain, as shown in Fig.4.

The additional jamming gain here is presented in the following two aspects:first,Decoy 1 and Decoy 2 can interfere with the radar independently or simultaneously,improving the probability of the jamming signal entering the radar seeker.Second, when multiple decoys jam simultaneously, if Decoy 1 fails to divert the missile from the ship, Decoy 2 may continue to jam, forming a multilevel defense system.

Let pidenote the probability of a jamming signal entering the radar seeker from Decoy i.The combinations of jamming signals when the two decoys work simultaneously are listed in Table 2.

Fig.4.Schematic of additional jamming gain from multiple decoys.

Table 2 Combinations of jamming signals and their probabilities of occurrence.

The symbol "√" indicates that the jamming signal has reached the radar seeker, and " × " indicates the opposite.For the 4th combination,neither Decoy 1 nor Decoy 2 works,and the effective jamming probability is 0.The effective jamming probability for other situations can be calculated as follows:

Owing to the incompatibility of the combinations, the total jamming probability is expressed as follows:

In Eqs.(18) and (19), piis an unknown element in practice and depends on many factors.Supposing that pi<1 and P(X) is a constant,it can be obtained that Pv′>Pv,which demonstrates the basic reliability of jamming cluster.For the formation optimization problem, the relationship between decoys is the focus.Here, Pv1represents the jamming effect of two decoys but Pv2and Pv3still refer to one decoy, for which we can set pi=1 and just Pv1is considered.

Similar to Refs.[10-13],the aforementioned analysis constrains the missile's incoming direction and only considers the scenario in which the missile is approaching horizontally.When the missile is coming from other directions, the jamming signal and the ship's echo signal are not within the same range gate of the radar seeker,as shown in Fig.5.In this case,the jamming does not work,and the angle will not be deceived.

To address this problem, a time delay can be added before the OAD forwards the radar signal.Since current monopulse radar seekers mostly use frequency agility technology [31], the decoy jamming signal can only be delayed rather than forwarded in advance.The missile's DOA is recorded as α, and the equivalent deployed distances of Decoy 1 and Decoy 2 can be expressed as follows:

According to Eq.(20), when the DOA of the missile is approximately 90°, the delayed jamming signal is close to the ship's echo signal,explaining the existence of the jamming-blindness area[14].This problem can be effectively solved if decoys are deployed in a two-dimensional (2D) plane, as shown in Fig.6.

Fig.5.Jamming scenario of decoys with straight line distribution for missiles from different directions.

In Fig.6,Decoy 2 can be used to interfere with the missile in the 90°direction.(x2,y2) denotes the position of Decoy 2, and the equivalent deployed distance can be calculated as follows:

Based on Eqs.(16), (18) and (21), we can calculate the effective jamming probability of a jamming formation composed of multiple decoys against incoming missiles from different distances and directions, and thus propose a method for assessing jamming formation optimization.

3.Optimization of USV-OAD cluster formation

For a given jamming formation, the effective jamming probability against a missile can be obtained based on previous theories.From a defense perspective, the key is to determine the optimal jamming formation, which is the focus of this subsection.

3.1.Jamming formation optimization model

The jamming formation optimization is designed to maximize the jamming probability while meeting the ship's defense requirements by deploying OADs appropriately.The defense requirements refer to the distance and DOA of the missile, which are determined in advance by the combat department based on the battlefield information, as shown below.

Fig.6.Jamming scenario of decoys deployed in a 2D plane.

In Fig.7, the jamming requirements are the critical jamming DOA and critical jamming range.The former indicates a highly probable direction for missiles to come from.The latter indicates the distance from which the missile is highly likely to start tracking.

The jamming requirements can be discretized.Hence,the set of jamming ranges is denoted as Rt={R1,R2,...,RA} and is used to calculate Eq.(17).The set of jamming DOAs is denoted as DOAs={α1,α2,...,αB}and is used to calculate Eq.(21).The positions of each decoy are (xi,yi) and i = 1,2,...,N.The effective jamming probability of the USV-OAD cluster formation is calculated as shown in Fig.8.

In Fig.8, the procedure mainly consists of three steps.

• First, combinations of multiple decoys are generated.Multiple decoys can be used for a single missile, which will either enhance or worsen the jamming effect.Assume that the maximum number of decoys required to jam a single missile is S,then there would be a total of M =ΣSs=1CsNcombinations.Each combination is denoted as mk.

• Second, the jamming probability of each combination is calculated for each missile distance and direction.The one with the maximum jamming probability is adopted.Based on the jamming requirements, A×B probability values are obtained.

• Third, we calculate the sum of all jamming probabilities for missiles coming from different directions and distances.The value is used to assess the jamming effectiveness of the formation, which can be expressed as

Fig.8.Process for calculating the jamming effectiveness of the USV-OAD cluster formation.

Eq.(22) represents the optimization objective: In terms of constraints, when the maximum defense distance and the beamwidth of the radar seeker are determined, the decoy will be deployed in a circular domain with a maximum radius of max(Rt)θHPBW.A penalty constraint is introduced to prevent the formation from having a jamming probability of zero for a considerable distance or DOA.pcriticaldenotes the minimum jamming probability.The formation could not be applied if the probability is lower than pcritical.In summary, the USV-OAD clusterformation optimization model can be expressed as follows:

Fig.7.Schematic of jamming requirements for USV-DOA cluster.

3.2.Improved PSO algorithm

Jamming formation optimization solves a complex multivariate maximum/minimum problem with a problem dimension twice the number of OADs.Based on the aforementioned analysis, the function mapping between the position to be optimized and the jamming effectiveness is nonlinear.Combination, filtering, and comparison of the jamming effects are involved.The problem cannot be solved using conventional function optimization methods.Intelligence optimization algorithms need to be used.

The PSO algorithm uses the spatial position of the particle as a solution to the problem, similar to the solution to the OADdeployed position optimization problem.Compared with other intelligent optimization algorithms such as genetic algorithm(GA),simulated annealing (SA) and ant colony optimization algorithm(ACO),PSO has higher flexibility and interpretability in solving the formation optimization problem.

3.2.1.Basics of the PSO algorithm

The PSO algorithm regards each feasible solution of a problem as a particle, and all particles are used to search for the optimal solution through the directional movement of the particles[32].The basic update formula for the PSO algorithm is expressed as follows:

The search capability of the PSO algorithm is related to its population size, initial location of the population and parameter settings.The calculation speed decreases as the population size increases.When using PSO, you get what you see.The optimal value determined is on the search path.For high-dimensional problems, the particle swarm exhibits a sparse distribution in the solution space, and the sparsity increases exponentially with the problem dimension,which limits the global search capability of the PSO algorithm,making it easy to miss the global optimum and fall into a local optimum.In this study, an increase in the number of OADs causes the problem dimension to increase linearly twice,thus limiting the performance of the PSO algorithm in determining the optimum.

3.2.2.Improvement strategies for the PSO algorithm

Inspired by biological behaviors, such as bird and fish swarms and the correspondingly derived swarm intelligence algorithms,while incorporating the mutation strategy of the GA and Metropolis criterion of simulated annealing(SA)[33],we propose a multi-tribe adaptive PSO algorithm with a mixed mutation strategy and Metropolis criterion (3M-APSO) to enhance the global and local search capabilities.

(1) Multiple-tribe strategy

In biological clusters, such as bird or fish swarms, when individuals learn experience from the population,they tend to draw experience from a limited number of objects, often those adjacent to them.As a result,the population becomes more diverse.In PSO,a new hierarchy called“tribe”can be formed between the swarm and individuals, and each particle belongs to a tribe.The updated formula of the particle speed can be expressed as

Fig.9.Search procedure for the optimal USV-OAD cluster formation.

where tr(i) indicates which tribe the ith particle belongs to, and Tbesttr(i)tis the historically optimal position of the tribe.The meaning of (c3,r3) is similar to those of (c1,r1) and (c2,r2).With tribes added, the population can maintain the characteristics of each tribe during the iterative update process, which considerably increases population diversity and suppresses the premature convergence of PSO in solving high-dimensional problems.

The adaptive inertia weight is used to improve the performance of the local search as follows:

where wmaxand wminare the maximum and minimum values,respectively,and tmaxdenotes the maximum number of iterations.

(2) Mutation strategy

In the GA, the gene mutation is performed when the parent generates the next generation to allow the children generation to have a strong search capability.The mutation operation can be introduced in the PSO algorithm by setting up a mutation probability operator, pmu=e-βt/tmax[34].After the conventional update of PSO, a random number between 0 and 1 is generated and compared with pmufor each particle.If the random number is smaller than pmu, two dimensions are randomly selected for each particle,and random initialization is conducted.Finally,the fitness value of the mutated particle is calculated.If it improves, the mutation operation will be kept;otherwise,the particle is cached,and whether the mutation is maintained or not is determined by applying the Metropolis criterion.

(3) Metropolis criterion

The Metropolis criterion is the core strategy of the SA algorithm.As a new solution is generated from an old one, the criterion probabilistically accepts it based on the fitness difference between the new and old solutions.The Metropolis criterion contains an additional parameter temperature T.The higher the temperature,the higher the acceptance probability,allowing the particle to have a strong global search capability at the beginning of the search.The Metropolis criterion calculates the probability of accepting the mutation of a particle as follows:

where fit(x) is the assessment function, and the probability is denoted as pa.

Subsection 3.1 states that the position of a particle in the solution space corresponds to the deployment positions of OADs.Therefore, the particle-based operations directly reflect the formation changes, which makes the introduced strategies specific and interpretable.Firstly, the multiple-tribe strategy increases the probability that the population is located in different regions during the iterative process, automatically maintaining a certain population diversity, i.e., the diversity of USV-OADs formation combinations.Secondly, the mutation strategy extends the search process of each particle to some other dimensions by changing the positions of some USV-OADs,which enhances the global searching ability and increases the probability of jumping out of the local optimum during the optimization search.Thirdly, the introduced Metropolis criterion puts a constraint on the mutation operations,which guarantees the directionality of the search process.To sum up, the improved PSO algorithm has the ability to handle highdimensional problems and is applicable to the formation optimization in this paper.

3.3.Implementation of the jamming formation optimization

Combining the aforementioned analysis with the USV-OAD cluster formation optimization model, the search procedure for the 3M-APSO algorithm optimization is constructed, as shown in Fig.9.

The sub-process in Fig.9 refers to the steps shown in Fig.8.The theoretical analysis and evaluation of the USV-OAD cluster are used as an external input to the formation optimization process,providing necessary computational modules to the algorithm.The improved strategies for the PSO algorithm add additional operations for each particle, and the pseudocode is as follows.

Algorithm 1 3M-APSO Algorithm

4.Simulation analysis

In this section, some practical jamming countermeasures are considered to validate the applicability of the jamming theorybased assessment method and the performance of the improved PSO algorithm to find the optimal formation.Given certain parameters, subsection 4.1 focuses on the jamming formation of the missile from a single direction, and subsection 4.2 focuses on the missile from different directions.

4.1.Single-directional defensive jamming formation

4.1.1.Single jamming range

The DOA of the missile is limited between -10°and 10°with a step size of 1°.The critical range of the missile is set to 20 km,indicating that the jamming range is single.The jamming and optimization parameters are as follows:

• θHPBW= 5°;

• Pt～U(1 kw，10 kw);

• Gtis the pattern function of the Gaussian beam [28];

• N∊{1,2,3};

• PjGj= 500 W;

• κ ～U(0.1,0.8);

• σ = 40 dBm2;

• pcritical= 0.02;

• The population of 3M-APSO:50;

• The number of tribes: 5;

• The maximum number of iterations:tmax= 200;

• wmax= 0.9, wmin= 0.2;

• The initial temperature:T =1000,attenuation coefficient:0.95;

• β = 0.1.

From the aforementioned parameters, we assume that the transmit power of the radar seeker and jamming loss of the jammer are uniformly distributed and random variables.When one, two,and three OADs are used,respectively,the performance of the APSO and 3M-APSO algorithms for each situation is as follows.

In Fig.10, the horizontal axis represents the number of iterations, and the vertical axis represents the fitness value calculated from Eq.(23).The legend reflects three different scenarios of formation optimization with respectively one decoy, two decoys and three decoys.It can be seen that the more decoys used, the better the jamming effect can be obtained.The similar fitness values of APSO and 3M-APSO demonstrate that when the number of decoys is small,which means that the dimension of the problem is low,the performance of 3M-APSO is approximated to APSO.However, the convergence rate of 3M-APSO is higher than that of APSO when more OADs are deployed.The results of the optimized OAD deployment positions are as follows (see Table 3).

The spatial distribution of the decoys and effective jamming probability distribution of the surrounding area are shown in Fig.11.

Fig.10.Iteration convergence curves of APSO and 3M-APSO for each jamming condition.

Table 3 Results of optimized deployment positions.

The scattered points in Fig.11 indicate the spatial positions of OADs, and the radius in polar coordinates indicates the effective jamming probability of the DOA.As shown, the decoy positions maximize the jamming probability for the missile from a single direction and exhibit symmetry on the top and bottom sides.Based on theoretical analysis,as every OAD has the same jamming effect,multiple OADs should be deployed centrally to fully utilize the advantages of power stacking, which validates the applicability of the algorithm.

4.1.2.Different jamming ranges

The radar seeker is usually turned on at 20 km from the target,and it takes a certain time for the seeker to lock on to the target.Therefore, Rtshould be within a specific range.Assuming Rt={10 km,12 km,...,20 km} and six OADs can be used, the iteration convergence curves are shown in Fig.12.

The other parameters are the same as those described in subsection 4.1.1.When six decoys are used, the dimension of the formation optimization problem is 12.From Fig.12, 3M-APSO performs better in finding the optimized solution in a highdimensional space.The optimized formation is shown in Fig.13.

In contrast to Fig.11, multiple decoys are distributed horizontally to jam the missile from different ranges.In Fig.11(a), decoys are placed on both sides of the ship.Based on the principle of angle deception jamming,the decoys below and above the protected ship cannot form a synergy, resulting in a waste of jamming resources.The formation obtained by the 3M-APSO algorithm is a distributed deployment of multiple decoys set on the same side,which shows a characteristic of centralization.The fitness values of the formations for missiles at different distances are shown in Fig.14.

The horizontal coordinate in Fig.14 indicates the jamming distance,whereas the vertical coordinate represents the fitness value for different jamming distances in the (-10°,10°) direction.From Eq.(17), a greater Rtindicates a higher effective jamming probability.The result obtained by APSO is slightly better than that of 3MAPSO only when Rt= 14 km.For other distances, the formation optimized by APSO is inferior to the search result of 3M-APSO,further demonstrating the advantages of the 3M-APSO algorithm in solving the USV-OAD formation-optimization problem.

4.2.Multi-directional defensive jamming formation

4.2.1.Single jamming range

Assuming that the missile comes from different directions, the DOAs are (-10°, 10°), (80°, 100°), (170°, 190°) and (260°, 280°),which can be noted as directions ①-④,respectively.If Rt=20 km and eight decoys exist,the search results of 3M-APSO are shown in Fig.15.

Based on the optimized formation,the decoys are distributed in groups of four, concentrated in area A near (500, 500) and area B near (-500, -500).Decoys placed in area A mainly interfere with the missile from directions ①and ②,and the projection distance of directions ①and ②is between 400 m and 600 m, which fully utilizes the advantage of concentrated jamming using multiple decoys.The decoys laid in area B are the same as those in area A.As shown in Fig.15(c),the optimized formation realizes the maximum jamming probability based on the missile's incoming direction.

Fig.11.Optimized formation and effective jamming probability distribution of the optimized results: (a) APSO; (b) 3M-APSO.

Fig.12.Iteration convergence curves of APSO and 3M-APSO for multiple jamming ranges.

Fig.14.Fitness values of the formation for different jamming ranges.

The iterative curve,optimized formation,and effective jamming probability distribution using 12 OADs when the probabilities of the missile approaching from all directions are equal are shown in Fig.16.

Fig.15.The search result of the 3M-APSO for the missile from different directions:(a) Iteration convergence curve;(b) Positions of OADs; (c)Effective jamming probability of the optimized formation.

Fig.16.Search results of 12 OADs for a missile from all directions: (a) Iteration convergence curve; (b) Positions of OADs; (c) Effective jamming probability of the optimized formation.

In Fig.16(b) and 16(c), the optimized formation is compared with a man-made formation in which all the decoys are uniformly distributed.The results of the man-made formation are marked by blue scatters and blue curves in Fig.16(b)and 16(c),respectively.In the optimized formation,the decoys are divided into four groups of three decoys each (4×3).These decoys are evenly placed around the ship in different directions, forming a cross-shaped pattern.Comparing Figs.15(c) and 16(c), we observe that the "cross" formation enhances the probability of effective jamming against four diagonal directions.Although a more uniform jamming effect can be obtained by the man-made evenly-distributed formation, the synergy between decoys is barely considered, and thus the jamming effect is inferior to that of the "cross" formation.

4.2.2.Different jamming ranges

When the probability of the missile coming from all directions is the same and the jamming range of the missile is Rt= {10 km,12 km,...,20 km}, the fitness value of the optimized formation in Fig.16(b)is 171.93 if pcritical=0.01.From Eq.(23),a positive value indicates that the formation has a blind zone for the jamming requirements.The application results of the 3M-APSO for this situation are shown in Fig.17.

From the convergence curve,a positive fitness indicates that the jamming formation is highly susceptible to the existence of jamming blindness to defense requirements, and the fitness value of the optimized formation is -12.23.When Fig.17(b) is compared with Fig.16(b), the optimized formation exhibits the same "cross"shape.The difference is that some of the decoys are set closer to the ship to jam closer missiles.As shown in Fig.17(c),placing the decoy close to the ship sacrifices the jamming ability when Rtis high yet contributes to the jamming effectiveness when Rtis low.The formation exhibits a net-like structure, also called a "crossing-net"shape.

Fig.17.Search results of 12 OADs for Rt ={10 km,12 km,...,20 km}and all directions:(a)Iteration convergence curve;(b) Positions of OADs; (c)Effective jamming probability of the optimized formation.

5.Conclusions

Cooperative engagement based on unmanned-platform is an inevitable trend of the future naval warfare in the digital age.Unlike traditional offboard active decoys in anti-missile operations, USVs that carry jammers can achieve a desired jamming effect through autonomous maneuvers.This study focused on the USV-OAD cluster formation, established the mathematical optimization model, and proposed a 3M-APSO algorithm.On the one hand, the proposed model delves into the functions of each individual in the cluster and their relationship.On the other hand, the 3M-APSO algorithm shows a better optimization performance in terms of convergence speed and global search capability.Unmanned USVOAD cluster could effectively overcome OADs' disadvantages like blind zones and could adapt to different formations for various defense purposes.How to utilize the USV-OADs is the key to making the cluster work,and the results of this paper can serve as theoretical and technical support.

In conclusion, different decoys in a USV-OAD cluster could be stacked, relayed, blind-filled, or conflicting.The stacking relationship means multiple decoys are placed close to each other and exhibit a superimposed jamming effect.The relay relationship refers to multiple decoys being distributed in the lateral distance to achieve a stepped jamming effect.The blind-fill is when multiple decoys are distributed in a certain direction to constitute the jamming area when jamming missiles with different DOAs.The conflicting relationship is that when multiple decoys jam a missile at the same time,some decoys cause the tracking beam to point at the ship due to an incorrect situation.The formation optimization model considers the aforementioned relationships, and three principles can be summarized,which are filling the jamming blind area,distributing the jammers along the range,and focusing energy during jamming.

The USV-OAD cluster formation is optimized to counter potential threats to the ship.In practice, the allocation of jamming resources when multiple missiles arrive simultaneously is another critical link for achieving the desired jamming effect.In the future,we will study the allocation of jamming resources for USV-OAD clusters based on the optimized formation and promote the use of clusters for anti-missile operations.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This paper is supported by the National Natural Science Foundation of China (Grant No.62101579).